In lithography, an exposure energy, such as ultraviolet light that is generated in an optical system, is passed from an aperture of the system through a mask (or reticle) and onto a target such as a silicon substrate. The mask typically may contain opaque and transparent regions formed in a predetermined pattern. The exposure energy exposes the mask pattern, thereby forming an aerial image of the mask. The aerial image is then used to form an image onto a layer of resist formed on the target. The resist is then developed for removing either the exposed portions of resist for a positive resist or the unexposed portions of resist for a negative resist. This forms a patterned substrate. A mask typically may comprise a transparent plate such as fused silica having opaque (chrome) elements on the plate used to define a pattern. A radiation source illuminates the mask according to well-known methods. The radiation transmitted through the mask and exposure tool projection optics forms a diffraction-limited latent image of the mask features on the photoresist. The patterned substrate can then be used in subsequent fabrication processes. In semiconductor manufacturing, such a patterned substrate can be used in deposition, etching, or ion implantation processes, to form integrated circuits having very small features.
In a manufacturing process using a lithographic projection apparatus, a pattern in a mask is imaged onto a substrate, which is at least partially covered by a layer of radiation-sensitive material (resist). Generally, lithographic patterning processes are understood by those who practice the profession. Information regarding exemplary processes may be obtained, for example, from the book xe2x80x9cMicrochip Fabrication: A Practical Guide to Semiconductor Processingxe2x80x9d, Third Edition, by Peter van Zant, McGraw Hill Publishing Co., 1997 ISBN 0-07-067250-4.
As the size of lithographically fabricated structures decreases, and the density of the structures increases, the cost and complexity of designing masks additionally increases. One method of reducing costs of lithographic fabrication is by optimizing the lithographic design with a lithographic simulation step prior to the actual manufacturing step. One specific method of lithographic simulation is drawn to simulating the aerial image of the mask. The aerial image is defined as an intensity distribution of light just prior to reaching the resist on a surface of a substrate, when the substrate is exposed via the mask in an exposure apparatus. In order to simulate the aerial image, a layout of a mask and exposure conditions (non-limiting examples include NA: Numerical Aperture, "sgr" (sigma): Partial Coherence Factor) of the lithographic apparatus are typically required as input parameters.
Lithographic apparatus may employ various types of projection radiation, non-limiting examples of which include light, ultra-violet (xe2x80x9cUVxe2x80x9d) radiation (including extreme UV (xe2x80x9cEUVxe2x80x9d), deep UV (xe2x80x9cDUVxe2x80x9d), and vacuum UV (xe2x80x9cVUVxe2x80x9d)), X-rays, ion beams or electron beams. The following have been considered exemplary exposure sources. Light may generally refer to certain mercury emissions, i.e., wavelengths of 550 nm for the f-line, 436 nm for the g-line, and 405 mn for the h-line. Near-UV or UV generally typically refer to other mercury emissions, i.e., 365 nm for the i-line. DUV generally refers to excimer laser emissions, such as KrF (248 nm) and ArF (193 nm). VUV may refer to excimer laser F2, i.e., 157 nm, Ar2, i.e., 126 nm, etc. EUV may refer to 10-15 nm. This last portion of the electromagnetic spectrum is very close to xe2x80x9csoft X-raysxe2x80x9d but has been named as xe2x80x9cEUVxe2x80x9d, possibly to avoid the bad reputation of X-ray patterning. Soft X-rays may refer to 1-15 nm, which may typically be used in X-ray lithography.
Depending on the type of radiation used and the particular design requirements of the apparatus, the projection system may be for example, refractive, reflective or catadioptric, and may comprise vitreous components, grazing-incidence mirrors, selective multi-layer coatings, magnetic and/or electrostatic field lenses, etc; for simplicity, such components may be loosely referred to in this text, either singly or collectively, as a xe2x80x9clensxe2x80x9d.
When the resist is exposed by the aerial image, there is an additional variable in that some of the exposure light is reflected back by the surface of the substrate, and then absorbed by the resist. Accordingly, not only the resist characteristics (regarding, for example, refractive index: Dill""s A,B,C) but also parameters regarding the characteristics of the substrate (e.g., refractive index) should be included as input parameters for simulating the latent image.
The so-called Hopkins model treats the electric field forming the image typically as a scalar and assumes the object being imaged is sufficiently thin so that its effect on the incident field is represented by a multiplicative function. It is advantageous to perform the image formation analysis in the Fourier domain (frequency space) in order to deal with the pupil function of the imaging system rather than the amplitude response function, and with the angular distribution or xe2x80x9ceffective sourcexe2x80x9d rather than with the mutual intensity.
There are several computer programs commercially available that calculate aerial images based on the Hopkins model. For example, the University of California at Berkeley, Department of Electrical Engineering and Computer Science, Berkeley, Calif., 94720, offers a program called SPLAT.
The Hopkins model is used to model the imaging of drawn design features under partially coherent illumination. A major problem in modeling aerial images under partially coherent illumination is the necessity to superimpose and add the effect of each individual illumination source that makes up the partially coherent source. In the Hopkins model, a two-dimensional by two-dimensional transmission cross-coefficient function (xe2x80x9cTCCxe2x80x9d) is pre-calculated, which captures all the effects of the lithographic projection apparatus, including NA, sigma, etc. As taught, for example in Born and Wolf, p. 603, once a TCC is known, systems with partially coherent illumination can be modeled by integrating the TCC over the Fourier transform of the transmission function for the geometrical layout feature under illumination.
Fundamentally, the TCC is a two-dimensional by two-dimensional correlation function with a continuous set of arguments. In practice, an assumption can be made that the feature patterns to be imaged are periodic in space. For such periodic patterns, the TCC has a large, but discrete, set of arguments. The TCC can then be represented as a matrix, with discrete columns and rows. For typical features interesting to the lithographer the size of this matrix is tremendous and restricts the scale and size of features that can be simulated. It is a purpose of any simulation algorithm based on the Hopkins model to reduce the size of this matrix by approximation, while retaining a reasonable degree of accuracy.
An exemplary projection lithography system is illustrated in FIGS. 1A and 1B. In FIG. 1A, light from illumination source 102 is focussed by condenser 104. The condensed light passes through the mask 106, then through the pupil 108 and onto the substrate 110. As shown in FIG. 1B, substrate 110 may comprise a top anti-reflective coating 112, a resist 118, a bottom anti-reflective coating 117, a top substrate layer 114 and a plurality of other substrate layers. As indicated in FIG. 1B, the focal plane may lie within the resist 118.
In the past, several numerical techniques have been applied to reduce the size of the TCC to reasonable scales. In one instance singular value decomposition has been applied to decompose the TCC into its eigenspectrum, sort the resulting eigenvectors in decreasing magnitude of their eigenvalues, and only retain a finite number of eigenvectors in order to approximate the TCC. An exemplary method for optical simulation for the system of FIG. 1A, that uses the Hopkins model, is illustrated in FIG. 2. As illustrated in FIG. 2, at step S202, the lithographic projection apparatus and mask parameters are input in the system. At step S204, the TCC is approximated by an eigenvector diagonalization. At step S206, a two-dimensional table of convolution integrals of the eigenfunctions with a discrete set of normalized rectangles is calculated and tabulated. The article, xe2x80x9cFast, Low-Complexity Mask Designxe2x80x9d, by N. Cobb et al., SPIE Vol. 2440, pgs. 313-326, the entire disclosure of which is incorporated by reference, teaches an exemplary method for completing step S206. Since the resulting table is on a discrete grid of possible layout-features, to attain an acceptable resolution, the table must be large. At step S208, since the TCC has been approximated, the illumination system can be modeled, i.e. the aerial image can be simulated by combining each two-dimensional convolution integral corresponding to each respective rectangle in a given proximity window.
This method has two drawbacks. First, an eigenvector diagonalization is a numerically expensive operation. Thus, for a user to change lithographic projection apparatus conditions, such as NA, sigma, illumination type or lens-aberration, an expensive recalculation of the TCC approximation that limits the use of the simulation tool is required. Second, the eigenspectrum of the TCC is a two-dimensional function. Representing the associated field vectors of the geometrical pattern under illumination requires a two-dimensional lookup table that limits the speed of the field calculation in the Hopkins algorithm. Specifically, since the field vectors are a two-dimensional table, two pointers are needed to access any respective address of data, which increases access time. Further, since the field vectors are a two-dimensional table, they are stored in a portion of the cache. Using a look-up table from a DRAM in a present day CPU is more expensive than performing multiplication because the cache miss penalty is larger than the time to perform the multiplication.
Accordingly, there is needed a method and apparatus for simulating a projection lithography system using the Hopkins model, that does not use a numerically expensive eigenvector diagonalization operation. What is further needed is a method and apparatus for simulating a projection lithography system using the Hopkins model that permits a user to change lithographic projection apparatus conditions, such as NA, sigma, illumination type or lens-aberration without requiring an expensive recalculation of the TCC approximation.
It is an object of the present invention to provide a method and apparatus for simulating a projection lithography system using the Hopkins model that does not use a numerically expensive eigenvector diagonalization operation.
It is another object of the present invention to provide a method and apparatus for simulating a projection lithography system using the Hopkins model that permits a user to change lithographic projection apparatus conditions without requiring an expensive recalculation of the TCC approximation.
It is still another object of the present invention to provide a method and apparatus for predicting the intensity field distribution (xe2x80x9caerial imagexe2x80x9d) at the surface and throughout the resist film of a substrate being irradiated in a lithographic imaging process.
In accordance with the foregoing objectives, the present invention writes the TCC as T(qxe2x80x2q), where q and qxe2x80x2 each are continuous frequencies in two-dimensional Fourier space, q=(qx, qy), qxe2x80x2=(qxe2x80x2x, qxe2x80x2y). The present invention approximates the TCC extremely well as a bilinear form of a basis function with kernel A[ij]. Furthermore, the present invention utilizes a set of orthogonal polynomials. The kernel A[ij] represents a small matrix that efficiently approximates the TCC for a broad range of illumination conditions used in modern lithographic processing. The present invention uses a lower number of arithmetic operations as a result of the use of a one-dimensional look-up table as opposed to the two-dimensional look-up table in the prior art. Furthermore, the one-dimensional look-up table achieves a higher cache hit ratio than a two-dimensional look-up table of the same resolution, thereby providing a more efficient system.
In general, in one embodiment, the invention features a method of simulating an aerial image projected from an optical system, the optical system including a pupil and a mask plane, the method comprising the steps of providing a mask to the mask plane, obtaining parameters for the optical system, calculating a kernel based on an orthogonal pupil projection of the parameters of the optical system onto a basis set, obtaining parameters of the mask, calculating a vector based on an orthogonal mask projection of the parameters of the mask onto a basis set, calculating a field intensity distribution using the kernel and the vector, and obtaining an aerial image from the field intensity distribution.
In one embodiment of the invention, the parameters for the optical system include aberrations.
In another embodiment of the invention, the step of calculating a kernel corresponding to the parameters of the optical system includes the step of generating a table of indefinite integrals based on a recurrence over a seed array of incomplete gamma functions.
In yet another embodiment of the invention, the step of calculating a kernel corresponding to the parameters of the optical system further includes the step of tabulating an array of orthogonal pupil projection coefficients corresponding to respective points in the pupil of the optical system, wherein the optical system is in-focus. More particularly, it further comprises the step of combining sample weights of an illuminator profile of the optical system with the array of orthogonal pupil projection coefficients.
In still another embodiment of the invention, the step of calculating a kernel corresponding to the parameters of the optical system further includes the step of tabulating an array of orthogonal pupil projection coefficients corresponding to respective points in the pupil of the optical system, wherein the optical system is either not in-focus or has aberrations. More particularly, it further comprises the step of combining sample weights of an illuminator profile of the optical system with the array of orthogonal pupil projection coefficients.
In still yet another embodiment of the invention, the step of calculating a kernel corresponding to the parameters of the optical system further includes the step of: tabulating an array of orthogonal pupil projection coefficients corresponding to respective points in the pupil of the optical system, wherein the optical system accounts for effects of diffusion of photoactive compounds in the resist. More particularly, it further comprises the step of combining sample weights of an illuminator profile of the optical system with the array of orthogonal pupil projection coefficients.
In a further embodiment of the invention, the step of calculating a vector corresponding to the parameters of the mask further includes the step of specifying a proximity window within the mask for geometric sampling.
In still a further embodiment of the invention, the step of calculating a vector corresponding to the parameters of the mask further includes the step of decomposing a geometric pattern of the mask into a disjoint set of rectangles and tabulating an array of projections of the rectangles within a proximity window. More particularly, the step of calculating a vector corresponding to the parameters of the mask further includes the step of correcting the array of projections of the rectangles based on the type of mask.
In general, in another aspect, the invention features a method for simulating an aerial image projected from an optical system, the optical system including a pupil and a mask plane, the method comprising the steps of: providing a mask to the mask plane, obtaining parameters for the optical system, obtaining parameters of the mask, projecting the components of electric field vectors to an orthogonal polynomial basis as described in further detail below, the polynomials including polynomials associated with the parameters of the optical system, and approximating the transmission cross-correlation function associated with the optical system based on the orthogonal projection of polynomials.
In general, in still another aspect, the invention features a simulation device operable to simulate an aerial image projected from an optical system, the optical system including a pupil and a mask plane, the simulation device comprising a first parameter obtainer for obtaining parameters of the optical system, a first calculator for calculating a kernel based on an orthogonal pupil projection of the parameters of the optical system onto a basis set, a second parameter obtainer for obtaining parameters of a mask provided to the mask plane, a second calculator for calculating a vector based on an orthogonal mask projection of the parameters of the mask onto a basis set, a third calculator for calculating a field intensity distribution using the kernel and the vector, and an aerial image obtainer for obtaining aerial image data from the field intensity distribution.
In one embodiment of the invention, the first, second, and third calculators are the same calculator.
In another embodiment of the invention, the parameters for the optical system include aberrations.
In yet another embodiment of the invention, the first calculator is operable to generate a table of indefinite integrals based on a fast recurrence over a seed array of incomplete gamma functions.
In still another embodiment of the invention, the first calculator is operable to tabulate an array of orthogonal pupil projection coefficients corresponding to respective points in the pupil of the optical system, wherein the optical system is in-focus.
In still yet another embodiment of the invention, the first calculator is operable to tabulate an array of orthogonal pupil projection coefficients corresponding to respective points in the pupil of the optical system, wherein the optical system is either not in-focus or has aberrations.
In a further embodiment of the invention, the first calculator is operable to tabulate an array of orthogonal pupil projection coefficients corresponding to respective points in the pupil of the optical system, wherein the optical system accounts for effects of photoactive compound diffusion in the resist.
In still a further embodiment of the invention, the second calculator is operable to specify a proximity window within the mask for geometric sampling.
In yet a further embodiment of the invention, the second calculator is operable to decompose a geometric pattern of the mask into a disjoint set of rectangles and tabulate an array of projections of the rectangles within a proximity window. More particularly the second calculator is operable to correct the array of projections of the rectangles based on the type of mask.
In general, in yet another aspect, the invention features a simulation device operable to simulate all aerial image projected from an optical system, the optical system including a pupil and a mask plane, the simulation device comprising a first parameter obtainer for obtaining parameters of the optical system, a second parameter obtainer for obtaining parameters of a mask provided to the mask plane, a first calculator for orthogonally projecting polynomials, the polynomials including polynomials associated with the parameters of the optical system, and a second calculator for approximating the transmission cross-coefficient associated with the optical system based on the orthogonal projection of polynomials.
Additional advantages of the present invention will become apparent to those skilled in the art from the following detailed description of exemplary embodiments of the present invention. The invention itself, together with further objects and advantages, can be better understood by reference to the following detailed description and the accompanying drawings.